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      "using Sigma\n",
      "loadvis()"
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     "language": "python",
     "metadata": {},
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    {
     "cell_type": "heading",
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     "source": [
      "Sampling Algorithms"
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     "cell_type": "markdown",
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      "Often all we want is a single sample or set of samples from a random variable $X$ conditioned on $Y$ being true.\n",
      "Additionally, other statistics such as conditional expectation, variance and conditional probability can be estimated from sets of samples.\n",
      "\n",
      "### Unconditional Samples\n",
      "\n",
      "In Sigma, to sample from $X$ unconditionally means to\n",
      "\n",
      "1. Sample an $\\omega \\in \\Omega$ according to $\\mathbb{P}$\n",
      "2. Evaluate $X(\\omega)$"
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = uniform(1,0.0,1.0)\n",
      "\u03c9 = [rand()]\n",
      "call(X,\u03c9)"
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     "language": "python",
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     "source": [
      "This is achieved more conviently simply using `rand(X)`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rand(X)"
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     "language": "python",
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      "Expectations of Real-valued random variables can be found using `sample_mean(X)`, which takes an additional keyword parameter `nsamples`. "
     ]
    },
    {
     "cell_type": "raw",
     "metadata": {},
     "source": [
      "sample_mean(X; nsamples = 1000)"
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    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Conditional Samples\n",
      "\n",
      "Conditionally sampling is a little more complex.\n",
      "To conditionally sample from $X$ given $Y$, simply call `rand(X,Y)`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = normal(0,1)\n",
      "Y = X > 0\n",
      "samples = rand(X,Y,1000)\n",
      "plot(x=samples,Geom.histogram)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
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     "cell_type": "markdown",
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     "source": [
      "`rand(X,Y)` is just a convenient synonym for `cond_sample_bfs(X,Y)` which draws exact samples. The alternative is `cond_sample_mh(X,Y)`.  We shall explain the difference between these two, for many it will suffice to know that `cond_sample_bfs` samples are _exact_, whereas `cond_sample_mh` using a variant of the Metropolis Hastings algorithm to handle higher dimensional problems, but as a result does not guarantee exact samples.\n",
      "To understand the differences, and how Sigma generates these samples we should first ask *what does it even mean to conditionaly sample?*\n",
      "\n",
      "#### Conditional Sampling Theory\n",
      "\n",
      "Recall $\\mathbb{P}:\\Sigma \\to [0,1]$ is our probability measure, i.e., a function which assigns probabilities to *subsets* of omega.  When we conditionally sampled from $X$ using `rand(X)` above, we really sampling an infinitesimly (but not zero) subset of.  To conditionally sample we first construct a new probability measure $\\mathbb{P_{Y}}$\n",
      "\n",
      "To construct a conditional query we first construct a conditional measure $\\mathbb{P}_{cond}$\n",
      "finds subsets of $\\Omega$.  Given a set of disjoint subsets, we can sample by sampling an abstraction.\n",
      "Intuitively this is correct because larger regions."
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     "source": [
      "### Abstract Metropolis Hastings"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
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     "source": [
      "## Making better proposals\n",
      "\n",
      "A major problem with abstract Metropolis Hastings is that a poor choice of where to split an abstraction can lead to rejected proposals.\n",
      "Consider the following preimage plot.  If we split down the middle two-ways to create four sub-boxes and use a uniform transition probability of 1/4 for each sub-box, the tiny preimage region in the upper-left hand corner will receive the same amout of proposal mass as each of the three other regions, which are all fully satisfiable.\n",
      "The result is that the small box (or some subset of it) will be proposed approximately 1/4 of the time and almost always rejected.\n",
      "\n",
      "This behaviour is a combination of two causes:\n",
      "1. our refinement strategy has no-backtracking, and hence if a sub-box is chosen, we will only further refine that subregion until a sat box is found\n",
      "2. an arbitrary splitting strategy will not in general split accoring to the actual underlying volumes.\n"
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      "X = uniform(0,1)\n",
      "Y = uniform(0,1)\n",
      "Z = (X > 0.5) | (Y < 0.5) | ((X \u2208 [0.25, 0.26]) & (Y \u2208 [0.75, 0.76]))\n",
      "plot_preimage(Z)"
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     "outputs": [
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       "text": [
        "Reached Max iterations - 1000\n"
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        "    // AMD support\n",
        "    if (typeof define === \"function\" && define.amd) {\n",
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        "        define(\"Gadfly\", [\"Snap.svg\"], function (Snap) {\n",
        "            return factory(Snap);\n",
        "        });\n",
        "    } else {\n",
        "        // Browser globals (glob is window)\n",
        "        // Snap adds itself to window\n",
        "        glob.Gadfly = factory(glob.Snap);\n",
        "    }\n",
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        "\n",
        "var Gadfly = {};\n",
        "\n",
        "// Get an x/y coordinate value in pixels\n",
        "var xPX = function(fig, x) {\n",
        "    var client_box = fig.node.getBoundingClientRect();\n",
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        "    return y * fig.node.viewBox.baseVal.height / client_box.height;\n",
        "};\n",
        "\n",
        "\n",
        "Snap.plugin(function (Snap, Element, Paper, global) {\n",
        "    // Traverse upwards from a snap element to find and return the first\n",
        "    // note with the \"plotroot\" class.\n",
        "    Element.prototype.plotroot = function () {\n",
        "        var element = this;\n",
        "        while (!element.hasClass(\"plotroot\") && element.parent() != null) {\n",
        "            element = element.parent();\n",
        "        }\n",
        "        return element;\n",
        "    };\n",
        "\n",
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        "        while (element.node.nodeName != \"svg\" && element.parent() != null) {\n",
        "            element = element.parent();\n",
        "        }\n",
        "        return element;\n",
        "    };\n",
        "\n",
        "    Element.prototype.plotbounds = function () {\n",
        "        var root = this.plotroot()\n",
        "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
        "        return {\n",
        "            x0: bbox.x,\n",
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        "            y0: bbox.y,\n",
        "            y1: bbox.y + bbox.height\n",
        "        };\n",
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        "\n",
        "    Element.prototype.plotcenter = function () {\n",
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        "        var bbox = root.select(\".guide.background\").node.getBBox();\n",
        "        return {\n",
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        "        };\n",
        "    };\n",
        "\n",
        "    // Emulate IE style mouseenter/mouseleave events, since Microsoft always\n",
        "    // does everything right.\n",
        "    // See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/\n",
        "    var events = [\"mouseenter\", \"mouseleave\"];\n",
        "\n",
        "    for (i in events) {\n",
        "        (function (event_name) {\n",
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        "                            return;\n",
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        "                        var reltg = event.relatedTarget ? event.relatedTarget :\n",
        "                            event.type == \"mouseout\" ? event.toElement : event.fromElement;\n",
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        "\n",
        "                        if (reltg != this.node) {\n",
        "                            return fn.apply(this, event);\n",
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        "\n",
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        "                    }\n",
        "                }\n",
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        "            };\n",
        "        })(events[i]);\n",
        "    }\n",
        "\n",
        "\n",
        "    Element.prototype.mousewheel = function (fn, scope) {\n",
        "        if (Snap.is(fn, \"function\")) {\n",
        "            var el = this;\n",
        "            var fn2 = function (event) {\n",
        "                fn.apply(el, [event]);\n",
        "            };\n",
        "        }\n",
        "\n",
        "        this.node.addEventListener(\n",
        "            /Firefox/i.test(navigator.userAgent) ? \"DOMMouseScroll\" : \"mousewheel\",\n",
        "            fn2);\n",
        "\n",
        "        return this;\n",
        "    };\n",
        "\n",
        "\n",
        "    // Snap's attr function can be too slow for things like panning/zooming.\n",
        "    // This is a function to directly update element attributes without going\n",
        "    // through eve.\n",
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        "            return this.node.getAttribute(key, val);\n",
        "        } else {\n",
        "            return this.node.setAttribute(key, val);\n",
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        "    };\n",
        "});\n",
        "\n",
        "\n",
        "// When the plot is moused over, emphasize the grid lines.\n",
        "Gadfly.plot_mouseover = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    init_pan_zoom(root);\n",
        "\n",
        "    var xgridlines = root.select(\".xgridlines\"),\n",
        "        ygridlines = root.select(\".ygridlines\");\n",
        "\n",
        "    xgridlines.data(\"unfocused_strokedash\",\n",
        "                    xgridlines.attr(\"stroke-dasharray\").replace(/px/g, \"mm\"))\n",
        "    ygridlines.data(\"unfocused_strokedash\",\n",
        "                    ygridlines.attr(\"stroke-dasharray\").replace(/px/g, \"mm\"))\n",
        "\n",
        "    // emphasize grid lines\n",
        "    var destcolor = root.data(\"focused_xgrid_color\");\n",
        "    xgridlines.attr(\"stroke-dasharray\", \"none\")\n",
        "              .selectAll(\"path\")\n",
        "              .animate({stroke: destcolor}, 250);\n",
        "\n",
        "    destcolor = root.data(\"focused_ygrid_color\");\n",
        "    ygridlines.attr(\"stroke-dasharray\", \"none\")\n",
        "              .selectAll(\"path\")\n",
        "              .animate({stroke: destcolor}, 250);\n",
        "\n",
        "    // reveal zoom slider\n",
        "    root.select(\".zoomslider\")\n",
        "        .animate({opacity: 1.0}, 250);\n",
        "};\n",
        "\n",
        "\n",
        "// Unemphasize grid lines on mouse out.\n",
        "Gadfly.plot_mouseout = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    var xgridlines = root.select(\".xgridlines\"),\n",
        "        ygridlines = root.select(\".ygridlines\");\n",
        "\n",
        "    var destcolor = root.data(\"unfocused_xgrid_color\");\n",
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        "    xgridlines.attr(\"stroke-dasharray\", xgridlines.data(\"unfocused_strokedash\"))\n",
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        "              .animate({stroke: destcolor}, 250);\n",
        "\n",
        "    destcolor = root.data(\"unfocused_ygrid_color\");\n",
        "    ygridlines.attr(\"stroke-dasharray\", ygridlines.data(\"unfocused_strokedash\"))\n",
        "              .selectAll(\"path\")\n",
        "              .animate({stroke: destcolor}, 250);\n",
        "\n",
        "    // hide zoom slider\n",
        "    root.select(\".zoomslider\")\n",
        "        .animate({opacity: 0.0}, 250);\n",
        "};\n",
        "\n",
        "\n",
        "var set_geometry_transform = function(root, tx, ty, scale) {\n",
        "    var xscalable = root.hasClass(\"xscalable\"),\n",
        "        yscalable = root.hasClass(\"yscalable\");\n",
        "\n",
        "    var old_scale = root.data(\"scale\");\n",
        "\n",
        "    var xscale = xscalable ? scale : 1.0,\n",
        "        yscale = yscalable ? scale : 1.0;\n",
        "\n",
        "    tx = xscalable ? tx : 0.0;\n",
        "    ty = yscalable ? ty : 0.0;\n",
        "\n",
        "    var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);\n",
        "\n",
        "    root.selectAll(\".geometry, image\")\n",
        "        .forEach(function (element, i) {\n",
        "            element.transform(t);\n",
        "        });\n",
        "\n",
        "    bounds = root.plotbounds();\n",
        "\n",
        "    if (yscalable) {\n",
        "        var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);\n",
        "        root.selectAll(\".xfixed\")\n",
        "            .forEach(function (element, i) {\n",
        "                element.transform(xfixed_t);\n",
        "            });\n",
        "\n",
        "        root.select(\".ylabels\")\n",
        "            .transform(xfixed_t)\n",
        "            .selectAll(\"text\")\n",
        "            .forEach(function (element, i) {\n",
        "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                    var cx = element.asPX(\"x\"),\n",
        "                        cy = element.asPX(\"y\");\n",
        "                    var st = element.data(\"static_transform\");\n",
        "                    unscale_t = new Snap.Matrix();\n",
        "                    unscale_t.scale(1, 1/scale, cx, cy).add(st);\n",
        "                    element.transform(unscale_t);\n",
        "\n",
        "                    var y = cy * scale + ty;\n",
        "                    element.attr(\"visibility\",\n",
        "                        bounds.y0 <= y && y <= bounds.y1 ? \"visible\" : \"hidden\");\n",
        "                }\n",
        "            });\n",
        "    }\n",
        "\n",
        "    if (xscalable) {\n",
        "        var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);\n",
        "        var xtrans = new Snap.Matrix().translate(tx, 0);\n",
        "        root.selectAll(\".yfixed\")\n",
        "            .forEach(function (element, i) {\n",
        "                element.transform(yfixed_t);\n",
        "            });\n",
        "\n",
        "        root.select(\".xlabels\")\n",
        "            .transform(yfixed_t)\n",
        "            .selectAll(\"text\")\n",
        "            .forEach(function (element, i) {\n",
        "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                    var cx = element.asPX(\"x\"),\n",
        "                        cy = element.asPX(\"y\");\n",
        "                    var st = element.data(\"static_transform\");\n",
        "                    unscale_t = new Snap.Matrix();\n",
        "                    unscale_t.scale(1/scale, 1, cx, cy).add(st);\n",
        "\n",
        "                    element.transform(unscale_t);\n",
        "\n",
        "                    var x = cx * scale + tx;\n",
        "                    element.attr(\"visibility\",\n",
        "                        bounds.x0 <= x && x <= bounds.x1 ? \"visible\" : \"hidden\");\n",
        "                    }\n",
        "            });\n",
        "    }\n",
        "\n",
        "    // we must unscale anything that is scale invariance: widths, raiduses, etc.\n",
        "    var size_attribs = [\"font-size\"];\n",
        "    var unscaled_selection = \".geometry, .geometry *\";\n",
        "    if (xscalable) {\n",
        "        size_attribs.push(\"rx\");\n",
        "        unscaled_selection += \", .xgridlines\";\n",
        "    }\n",
        "    if (yscalable) {\n",
        "        size_attribs.push(\"ry\");\n",
        "        unscaled_selection += \", .ygridlines\";\n",
        "    }\n",
        "\n",
        "    root.selectAll(unscaled_selection)\n",
        "        .forEach(function (element, i) {\n",
        "            // circle need special help\n",
        "            if (element.node.nodeName == \"circle\") {\n",
        "                var cx = element.attribute(\"cx\"),\n",
        "                    cy = element.attribute(\"cy\");\n",
        "                unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale,\n",
        "                                                        cx, cy);\n",
        "                element.transform(unscale_t);\n",
        "                return;\n",
        "            }\n",
        "\n",
        "            for (i in size_attribs) {\n",
        "                var key = size_attribs[i];\n",
        "                var val = parseFloat(element.attribute(key));\n",
        "                if (val !== undefined && val != 0 && !isNaN(val)) {\n",
        "                    element.attribute(key, val * old_scale / scale);\n",
        "                }\n",
        "            }\n",
        "        });\n",
        "};\n",
        "\n",
        "\n",
        "// Find the most appropriate tick scale and update label visibility.\n",
        "var update_tickscale = function(root, scale, axis) {\n",
        "    if (!root.hasClass(axis + \"scalable\")) return;\n",
        "\n",
        "    var tickscales = root.data(axis + \"tickscales\");\n",
        "    var best_tickscale = 1.0;\n",
        "    var best_tickscale_dist = Infinity;\n",
        "    for (tickscale in tickscales) {\n",
        "        var dist = Math.abs(Math.log(tickscale) - Math.log(scale));\n",
        "        if (dist < best_tickscale_dist) {\n",
        "            best_tickscale_dist = dist;\n",
        "            best_tickscale = tickscale;\n",
        "        }\n",
        "    }\n",
        "\n",
        "    if (best_tickscale != root.data(axis + \"tickscale\")) {\n",
        "        root.data(axis + \"tickscale\", best_tickscale);\n",
        "        var mark_inscale_gridlines = function (element, i) {\n",
        "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
        "            element.attribute(\"gadfly:inscale\", inscale);\n",
        "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
        "        };\n",
        "\n",
        "        var mark_inscale_labels = function (element, i) {\n",
        "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
        "            element.attribute(\"gadfly:inscale\", inscale);\n",
        "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
        "        };\n",
        "\n",
        "        root.select(\".\" + axis + \"gridlines\").selectAll(\"path\").forEach(mark_inscale_gridlines);\n",
        "        root.select(\".\" + axis + \"labels\").selectAll(\"text\").forEach(mark_inscale_labels);\n",
        "    }\n",
        "};\n",
        "\n",
        "\n",
        "var set_plot_pan_zoom = function(root, tx, ty, scale) {\n",
        "    var old_scale = root.data(\"scale\");\n",
        "    var bounds = root.plotbounds();\n",
        "\n",
        "    var width = bounds.x1 - bounds.x0,\n",
        "        height = bounds.y1 - bounds.y0;\n",
        "\n",
        "    // compute the viewport derived from tx, ty, and scale\n",
        "    var x_min = -width * scale - (scale * width - width),\n",
        "        x_max = width * scale,\n",
        "        y_min = -height * scale - (scale * height - height),\n",
        "        y_max = height * scale;\n",
        "\n",
        "    var x0 = bounds.x0 - scale * bounds.x0,\n",
        "        y0 = bounds.y0 - scale * bounds.y0;\n",
        "\n",
        "    var tx = Math.max(Math.min(tx - x0, x_max), x_min),\n",
        "        ty = Math.max(Math.min(ty - y0, y_max), y_min);\n",
        "\n",
        "    tx += x0;\n",
        "    ty += y0;\n",
        "\n",
        "    // when the scale change, we may need to alter which set of\n",
        "    // ticks is being displayed\n",
        "    if (scale != old_scale) {\n",
        "        update_tickscale(root, scale, \"x\");\n",
        "        update_tickscale(root, scale, \"y\");\n",
        "    }\n",
        "\n",
        "    set_geometry_transform(root, tx, ty, scale);\n",
        "\n",
        "    root.data(\"scale\", scale);\n",
        "    root.data(\"tx\", tx);\n",
        "    root.data(\"ty\", ty);\n",
        "};\n",
        "\n",
        "\n",
        "var scale_centered_translation = function(root, scale) {\n",
        "    var bounds = root.plotbounds();\n",
        "\n",
        "    var width = bounds.x1 - bounds.x0,\n",
        "        height = bounds.y1 - bounds.y0;\n",
        "\n",
        "    var tx0 = root.data(\"tx\"),\n",
        "        ty0 = root.data(\"ty\");\n",
        "\n",
        "    var scale0 = root.data(\"scale\");\n",
        "\n",
        "    // how off from center the current view is\n",
        "    var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2),\n",
        "        yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2);\n",
        "\n",
        "    // rescale offsets\n",
        "    xoff = xoff * scale / scale0;\n",
        "    yoff = yoff * scale / scale0;\n",
        "\n",
        "    // adjust for the panel position being scaled\n",
        "    var x_edge_adjust = bounds.x0 * (1 - scale),\n",
        "        y_edge_adjust = bounds.y0 * (1 - scale);\n",
        "\n",
        "    return {\n",
        "        x: xoff + x_edge_adjust + (width - width * scale) / 2,\n",
        "        y: yoff + y_edge_adjust + (height - height * scale) / 2\n",
        "    };\n",
        "};\n",
        "\n",
        "\n",
        "// Initialize data for panning zooming if it isn't already.\n",
        "var init_pan_zoom = function(root) {\n",
        "    if (root.data(\"zoompan-ready\")) {\n",
        "        return;\n",
        "    }\n",
        "\n",
        "    // The non-scaling-stroke trick. Rather than try to correct for the\n",
        "    // stroke-width when zooming, we force it to a fixed value.\n",
        "    var px_per_mm = root.node.getCTM().a;\n",
        "\n",
        "    // Drag events report deltas in pixels, which we'd like to convert to\n",
        "    // millimeters.\n",
        "    root.data(\"px_per_mm\", px_per_mm);\n",
        "\n",
        "    root.selectAll(\"path\")\n",
        "        .forEach(function (element, i) {\n",
        "        sw = element.asPX(\"stroke-width\") * px_per_mm;\n",
        "        if (sw > 0) {\n",
        "            element.attribute(\"stroke-width\", sw);\n",
        "            element.attribute(\"vector-effect\", \"non-scaling-stroke\");\n",
        "        }\n",
        "    });\n",
        "\n",
        "    // Store ticks labels original tranformation\n",
        "    root.selectAll(\".xlabels > text, .ylabels > text\")\n",
        "        .forEach(function (element, i) {\n",
        "            var lm = element.transform().localMatrix;\n",
        "            element.data(\"static_transform\",\n",
        "                new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));\n",
        "        });\n",
        "\n",
        "    if (root.data(\"tx\") === undefined) root.data(\"tx\", 0);\n",
        "    if (root.data(\"ty\") === undefined) root.data(\"ty\", 0);\n",
        "    if (root.data(\"scale\") === undefined) root.data(\"scale\", 1.0);\n",
        "    if (root.data(\"xtickscales\") === undefined) {\n",
        "\n",
        "        // index all the tick scales that are listed\n",
        "        var xtickscales = {};\n",
        "        var ytickscales = {};\n",
        "        var add_x_tick_scales = function (element, i) {\n",
        "            xtickscales[element.attribute(\"gadfly:scale\")] = true;\n",
        "        };\n",
        "        var add_y_tick_scales = function (element, i) {\n",
        "            ytickscales[element.attribute(\"gadfly:scale\")] = true;\n",
        "        };\n",
        "\n",
        "        root.select(\".xgridlines\").selectAll(\"path\").forEach(add_x_tick_scales);\n",
        "        root.select(\".ygridlines\").selectAll(\"path\").forEach(add_y_tick_scales);\n",
        "        root.select(\".xlabels\").selectAll(\"text\").forEach(add_x_tick_scales);\n",
        "        root.select(\".ylabels\").selectAll(\"text\").forEach(add_y_tick_scales);\n",
        "\n",
        "        root.data(\"xtickscales\", xtickscales);\n",
        "        root.data(\"ytickscales\", ytickscales);\n",
        "        root.data(\"xtickscale\", 1.0);\n",
        "    }\n",
        "\n",
        "    var min_scale = 1.0, max_scale = 1.0;\n",
        "    for (scale in xtickscales) {\n",
        "        min_scale = Math.min(min_scale, scale);\n",
        "        max_scale = Math.max(max_scale, scale);\n",
        "    }\n",
        "    for (scale in ytickscales) {\n",
        "        min_scale = Math.min(min_scale, scale);\n",
        "        max_scale = Math.max(max_scale, scale);\n",
        "    }\n",
        "    root.data(\"min_scale\", min_scale);\n",
        "    root.data(\"max_scale\", max_scale);\n",
        "\n",
        "    // store the original positions of labels\n",
        "    root.select(\".xlabels\")\n",
        "        .selectAll(\"text\")\n",
        "        .forEach(function (element, i) {\n",
        "            element.data(\"x\", element.asPX(\"x\"));\n",
        "        });\n",
        "\n",
        "    root.select(\".ylabels\")\n",
        "        .selectAll(\"text\")\n",
        "        .forEach(function (element, i) {\n",
        "            element.data(\"y\", element.asPX(\"y\"));\n",
        "        });\n",
        "\n",
        "    // mark grid lines and ticks as in or out of scale.\n",
        "    var mark_inscale = function (element, i) {\n",
        "        element.attribute(\"gadfly:inscale\", element.attribute(\"gadfly:scale\") == 1.0);\n",
        "    };\n",
        "\n",
        "    root.select(\".xgridlines\").selectAll(\"path\").forEach(mark_inscale);\n",
        "    root.select(\".ygridlines\").selectAll(\"path\").forEach(mark_inscale);\n",
        "    root.select(\".xlabels\").selectAll(\"text\").forEach(mark_inscale);\n",
        "    root.select(\".ylabels\").selectAll(\"text\").forEach(mark_inscale);\n",
        "\n",
        "    // figure out the upper ond lower bounds on panning using the maximum\n",
        "    // and minum grid lines\n",
        "    var bounds = root.plotbounds();\n",
        "    var pan_bounds = {\n",
        "        x0: 0.0,\n",
        "        y0: 0.0,\n",
        "        x1: 0.0,\n",
        "        y1: 0.0\n",
        "    };\n",
        "\n",
        "    root.select(\".xgridlines\")\n",
        "        .selectAll(\"path\")\n",
        "        .forEach(function (element, i) {\n",
        "            if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                var bbox = element.node.getBBox();\n",
        "                if (bounds.x1 - bbox.x < pan_bounds.x0) {\n",
        "                    pan_bounds.x0 = bounds.x1 - bbox.x;\n",
        "                }\n",
        "                if (bounds.x0 - bbox.x > pan_bounds.x1) {\n",
        "                    pan_bounds.x1 = bounds.x0 - bbox.x;\n",
        "                }\n",
        "            }\n",
        "        });\n",
        "\n",
        "    root.select(\".ygridlines\")\n",
        "        .selectAll(\"path\")\n",
        "        .forEach(function (element, i) {\n",
        "            if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                var bbox = element.node.getBBox();\n",
        "                if (bounds.y1 - bbox.y < pan_bounds.y0) {\n",
        "                    pan_bounds.y0 = bounds.y1 - bbox.y;\n",
        "                }\n",
        "                if (bounds.y0 - bbox.y > pan_bounds.y1) {\n",
        "                    pan_bounds.y1 = bounds.y0 - bbox.y;\n",
        "                }\n",
        "            }\n",
        "        });\n",
        "\n",
        "    // nudge these values a little\n",
        "    pan_bounds.x0 -= 5;\n",
        "    pan_bounds.x1 += 5;\n",
        "    pan_bounds.y0 -= 5;\n",
        "    pan_bounds.y1 += 5;\n",
        "    root.data(\"pan_bounds\", pan_bounds);\n",
        "\n",
        "    // Set all grid lines at scale 1.0 to visible. Out of bounds lines\n",
        "    // will be clipped.\n",
        "    root.select(\".xgridlines\")\n",
        "        .selectAll(\"path\")\n",
        "        .forEach(function (element, i) {\n",
        "            if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                element.attr(\"visibility\", \"visible\");\n",
        "            }\n",
        "        });\n",
        "\n",
        "    root.select(\".ygridlines\")\n",
        "        .selectAll(\"path\")\n",
        "        .forEach(function (element, i) {\n",
        "            if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
        "                element.attr(\"visibility\", \"visible\");\n",
        "            }\n",
        "        });\n",
        "\n",
        "    root.data(\"zoompan-ready\", true)\n",
        "};\n",
        "\n",
        "\n",
        "// Panning\n",
        "Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {\n",
        "    var root = this.plotroot();\n",
        "    var px_per_mm = root.data(\"px_per_mm\");\n",
        "    dx /= px_per_mm;\n",
        "    dy /= px_per_mm;\n",
        "\n",
        "    var tx0 = root.data(\"tx\"),\n",
        "        ty0 = root.data(\"ty\");\n",
        "\n",
        "    var dx0 = root.data(\"dx\"),\n",
        "        dy0 = root.data(\"dy\");\n",
        "\n",
        "    root.data(\"dx\", dx);\n",
        "    root.data(\"dy\", dy);\n",
        "\n",
        "    dx = dx - dx0;\n",
        "    dy = dy - dy0;\n",
        "\n",
        "    var tx = tx0 + dx,\n",
        "        ty = ty0 + dy;\n",
        "\n",
        "    set_plot_pan_zoom(root, tx, ty, root.data(\"scale\"));\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.guide_background_drag_onstart = function(x, y, event) {\n",
        "    var root = this.plotroot();\n",
        "    root.data(\"dx\", 0);\n",
        "    root.data(\"dy\", 0);\n",
        "    init_pan_zoom(root);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.guide_background_drag_onend = function(event) {\n",
        "    var root = this.plotroot();\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.guide_background_scroll = function(event) {\n",
        "    if (event.shiftKey) {\n",
        "        var root = this.plotroot();\n",
        "        init_pan_zoom(root);\n",
        "        var new_scale = root.data(\"scale\") * Math.pow(2, 0.002 * event.wheelDelta);\n",
        "        new_scale = Math.max(\n",
        "            root.data(\"min_scale\"),\n",
        "            Math.min(root.data(\"max_scale\"), new_scale))\n",
        "        update_plot_scale(root, new_scale);\n",
        "        event.stopPropagation();\n",
        "    }\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_button_mouseover = function(event) {\n",
        "    this.select(\".button_logo\")\n",
        "         .animate({fill: this.data(\"mouseover_color\")}, 100);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_button_mouseout = function(event) {\n",
        "     this.select(\".button_logo\")\n",
        "         .animate({fill: this.data(\"mouseout_color\")}, 100);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_zoomout_click = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    init_pan_zoom(root);\n",
        "    var min_scale = root.data(\"min_scale\"),\n",
        "        scale = root.data(\"scale\");\n",
        "    Snap.animate(\n",
        "        scale,\n",
        "        Math.max(min_scale, scale / 1.5),\n",
        "        function (new_scale) {\n",
        "            update_plot_scale(root, new_scale);\n",
        "        },\n",
        "        200);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_zoomin_click = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    init_pan_zoom(root);\n",
        "    var max_scale = root.data(\"max_scale\"),\n",
        "        scale = root.data(\"scale\");\n",
        "\n",
        "    Snap.animate(\n",
        "        scale,\n",
        "        Math.min(max_scale, scale * 1.5),\n",
        "        function (new_scale) {\n",
        "            update_plot_scale(root, new_scale);\n",
        "        },\n",
        "        200);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_track_click = function(event) {\n",
        "    // TODO\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_thumb_mousedown = function(event) {\n",
        "    this.animate({fill: this.data(\"mouseover_color\")}, 100);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_thumb_mouseup = function(event) {\n",
        "    this.animate({fill: this.data(\"mouseout_color\")}, 100);\n",
        "};\n",
        "\n",
        "\n",
        "// compute the position in [0, 1] of the zoom slider thumb from the current scale\n",
        "var slider_position_from_scale = function(scale, min_scale, max_scale) {\n",
        "    if (scale >= 1.0) {\n",
        "        return 0.5 + 0.5 * (Math.log(scale) / Math.log(max_scale));\n",
        "    }\n",
        "    else {\n",
        "        return 0.5 * (Math.log(scale) - Math.log(min_scale)) / (0 - Math.log(min_scale));\n",
        "    }\n",
        "}\n",
        "\n",
        "\n",
        "var update_plot_scale = function(root, new_scale) {\n",
        "    var trans = scale_centered_translation(root, new_scale);\n",
        "    set_plot_pan_zoom(root, trans.x, trans.y, new_scale);\n",
        "\n",
        "    root.selectAll(\".zoomslider_thumb\")\n",
        "        .forEach(function (element, i) {\n",
        "            var min_pos = element.data(\"min_pos\"),\n",
        "                max_pos = element.data(\"max_pos\"),\n",
        "                min_scale = root.data(\"min_scale\"),\n",
        "                max_scale = root.data(\"max_scale\");\n",
        "            var xmid = (min_pos + max_pos) / 2;\n",
        "            var xpos = slider_position_from_scale(new_scale, min_scale, max_scale);\n",
        "            element.transform(new Snap.Matrix().translate(\n",
        "                Math.max(min_pos, Math.min(\n",
        "                         max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0));\n",
        "    });\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y) {\n",
        "    var root = this.plotroot();\n",
        "    var min_pos = this.data(\"min_pos\"),\n",
        "        max_pos = this.data(\"max_pos\"),\n",
        "        min_scale = root.data(\"min_scale\"),\n",
        "        max_scale = root.data(\"max_scale\"),\n",
        "        old_scale = root.data(\"old_scale\");\n",
        "\n",
        "    var px_per_mm = root.data(\"px_per_mm\");\n",
        "    dx /= px_per_mm;\n",
        "    dy /= px_per_mm;\n",
        "\n",
        "    var xmid = (min_pos + max_pos) / 2;\n",
        "    var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) +\n",
        "                   dx / (max_pos - min_pos);\n",
        "\n",
        "    // compute the new scale\n",
        "    var new_scale;\n",
        "    if (xpos >= 0.5) {\n",
        "        new_scale = Math.exp(2.0 * (xpos - 0.5) * Math.log(max_scale));\n",
        "    }\n",
        "    else {\n",
        "        new_scale = Math.exp(2.0 * xpos * (0 - Math.log(min_scale)) +\n",
        "                        Math.log(min_scale));\n",
        "    }\n",
        "    new_scale = Math.min(max_scale, Math.max(min_scale, new_scale));\n",
        "\n",
        "    update_plot_scale(root, new_scale);\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_thumb_dragstart = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    init_pan_zoom(root);\n",
        "\n",
        "    // keep track of what the scale was when we started dragging\n",
        "    root.data(\"old_scale\", root.data(\"scale\"));\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.zoomslider_thumb_dragend = function(event) {\n",
        "};\n",
        "\n",
        "\n",
        "var toggle_color_class = function(root, color_class, ison) {\n",
        "    var guides = root.selectAll(\".guide.\" + color_class + \",.guide .\" + color_class);\n",
        "    var geoms = root.selectAll(\".geometry.\" + color_class + \",.geometry .\" + color_class);\n",
        "    if (ison) {\n",
        "        guides.animate({opacity: 0.5}, 250);\n",
        "        geoms.animate({opacity: 0.0}, 250);\n",
        "    } else {\n",
        "        guides.animate({opacity: 1.0}, 250);\n",
        "        geoms.animate({opacity: 1.0}, 250);\n",
        "    }\n",
        "};\n",
        "\n",
        "\n",
        "Gadfly.colorkey_swatch_click = function(event) {\n",
        "    var root = this.plotroot();\n",
        "    var color_class = this.data(\"color_class\");\n",
        "\n",
        "    if (event.shiftKey) {\n",
        "        root.selectAll(\".colorkey text\")\n",
        "            .forEach(function (element) {\n",
        "                var other_color_class = element.data(\"color_class\");\n",
        "                if (other_color_class != color_class) {\n",
        "                    toggle_color_class(root, other_color_class,\n",
        "                                       element.attr(\"opacity\") == 1.0);\n",
        "                }\n",
        "            });\n",
        "    } else {\n",
        "        toggle_color_class(root, color_class, this.attr(\"opacity\") == 1.0);\n",
        "    }\n",
        "};\n",
        "\n",
        "\n",
        "return Gadfly;\n",
        "\n",
        "}));\n",
        "\n",
        "\n",
        "//@ sourceURL=gadfly.js\n",
        "\n",
        "(function (glob, factory) {\n",
        "    // AMD support\n",
        "      if (typeof require === \"function\" && typeof define === \"function\" && define.amd) {\n",
        "        require([\"Snap.svg\", \"Gadfly\"], function (Snap, Gadfly) {\n",
        "            factory(Snap, Gadfly);\n",
        "        });\n",
        "      } else {\n",
        "          factory(glob.Snap, glob.Gadfly);\n",
        "      }\n",
        "})(window, function (Snap, Gadfly) {\n",
        "    var fig = Snap(\"#fig-c15b0f9c69384aca8dd9630932aa77e5\");\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-10\")\n",
        "   .mouseenter(Gadfly.plot_mouseover)\n",
        ".mouseleave(Gadfly.plot_mouseout)\n",
        ".mousewheel(Gadfly.guide_background_scroll)\n",
        ".drag(Gadfly.guide_background_drag_onmove,\n",
        "      Gadfly.guide_background_drag_onstart,\n",
        "      Gadfly.guide_background_drag_onend)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-13\")\n",
        "   .plotroot().data(\"unfocused_ygrid_color\", \"#D0D0E0\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-13\")\n",
        "   .plotroot().data(\"focused_ygrid_color\", \"#A0A0A0\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-14\")\n",
        "   .plotroot().data(\"unfocused_xgrid_color\", \"#D0D0E0\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-14\")\n",
        "   .plotroot().data(\"focused_xgrid_color\", \"#A0A0A0\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-18\")\n",
        "   .data(\"mouseover_color\", \"#cd5c5c\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-18\")\n",
        "   .data(\"mouseout_color\", \"#6a6a6a\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-18\")\n",
        "   .click(Gadfly.zoomslider_zoomin_click)\n",
        ".mouseenter(Gadfly.zoomslider_button_mouseover)\n",
        ".mouseleave(Gadfly.zoomslider_button_mouseout)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-20\")\n",
        "   .data(\"max_pos\", 95.88)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-20\")\n",
        "   .data(\"min_pos\", 78.88)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-20\")\n",
        "   .click(Gadfly.zoomslider_track_click);\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-21\")\n",
        "   .data(\"max_pos\", 95.88)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-21\")\n",
        "   .data(\"min_pos\", 78.88)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-21\")\n",
        "   .data(\"mouseover_color\", \"#cd5c5c\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-21\")\n",
        "   .data(\"mouseout_color\", \"#6a6a6a\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-21\")\n",
        "   .drag(Gadfly.zoomslider_thumb_dragmove,\n",
        "     Gadfly.zoomslider_thumb_dragstart,\n",
        "     Gadfly.zoomslider_thumb_dragend)\n",
        ".mousedown(Gadfly.zoomslider_thumb_mousedown)\n",
        ".mouseup(Gadfly.zoomslider_thumb_mouseup)\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-22\")\n",
        "   .data(\"mouseover_color\", \"#cd5c5c\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-22\")\n",
        "   .data(\"mouseout_color\", \"#6a6a6a\")\n",
        ";\n",
        "fig.select(\"#fig-c15b0f9c69384aca8dd9630932aa77e5-element-22\")\n",
        "   .click(Gadfly.zoomslider_zoomout_click)\n",
        ".mouseenter(Gadfly.zoomslider_button_mouseover)\n",
        ".mouseleave(Gadfly.zoomslider_button_mouseout)\n",
        ";\n",
        "    });\n",
        "]]> </script>\n",
        "</svg>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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        "Plot(...)"
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The fundamental problem is found in all MCMC algorithms, our proposal distribution poorly reflects the distribution of interest.  In our case it can be off by several (10s or 100s) orders of magnitude.\n",
      "We first address this with methods which are non-adaptive, that is, the proposal distribution does not change.\n",
      "Later we can conider *adaptive* methods, which have addition constraints to ensure correctness.\n",
      "\n",
      "### Estimating Relative Volumes\n",
      "\n",
      "Consider a subdivision of an abstraction into sub-abstractions $\\tilde{A}$ and $\\tilde{B}$, a good proposal distribution will assign mass to these according to the volume of the true preimage, each subatraction subsumes.  This brings a circularity; since if we knew the volumes within each subregion there would be no need for metropolis hastings at all."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Bounding box as volume statistic\n",
      "\n",
      "The idea here is just to find the a bounding box.  We can do this using a binary search on each dimension individually"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Refine each sub-abstraction further to get an estimate of relative volumes\n",
      "\n",
      "Refine each box a few times and see how many boxes we can remove"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### RRT as a volume statistic\n",
      "\n",
      "Rapidly exploring random trees (RRT) and its variants are sample based methods which work well for motion planning in high dimensional spaces.  The fact that they are able to fill high dimensional spaces with a tree is motivation for their use in this case.\n",
      "\n",
      "Basic algorithm:\n",
      "\n",
      "1. Sample a point in the coordinate space\n",
      "2. Find the nearest point in the current tree to that point\n",
      "3. Generate a node as close to point found in 1 as possible under constraint that an edge between it and node from 2 can be drawn.\n",
      "\n",
      "The challenges in adapting this method to our needs are that\n",
      "\n",
      "1. RRT relies on being able to draw points uniformly over the space. The spaces used in motion planning have feasible regions large enough that simple rejection sampling is good enough to get points. This property does not hold for us, where the feasible regions are likely tiny.\n",
      "\n",
      "2. RRT is primarily concerned with expansion into continuous regions.  If there are discontinuities in the state space, they are not of importance because it means the robot can not possible reach the target.\n",
      "\n",
      "3. When a new node is sampled, an edge is grown from the closest node in the tree as close as possible to it.  This requires that we know that the edge is not interesecting a non disjoint region.  I'm not entirely sure how they check this, but for us it could be done with an SMT solver, but that seems expensive.  The goal is to have a fast approximation.\n",
      "\n",
      "- Whether a line in the state space collides with the unfeasible region can be framed as an SMT problem. Does there exist a point on this line which is in the unfeasible regions.\n",
      "- Alternatively, one could just check points.  Checking points is easy.  One could also just check a lot of points along the line.\n",
      "\n",
      "Heuristic to try:\n",
      "\n",
      "When splitting a box A into $A_i,..,A_n$.  For each $A_i$\n",
      "\n",
      "1. Sample a valid point using SMT or abstract interpretation\n",
      "2. Construct an nball or ncube of radius $r$ around it truncated to its subregion and sample within that: $x_rand$\n",
      "3. Find nearest node in tree to $x_rand$\n",
      "4. Grow node as near as possible to $x_rand$ using either SMT or simply sampling along line to ensure its fully contained.\n",
      "5. Periodically, try to find point outside of convex hull of tree.\n"
     ]
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 ]
}